\section{Passive control}

\begin{frame}{\thesection.\ \insertsection}
\begin{block}{Passive control}
    Makes use of the spacecraft’s natural dynamics to ensure a stable equilibrium at the desired attitude, and is useful when accuracy requirements are coarse.
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The stability of a system refers to the asymptotic behavior of the system (as \( t \to \infty \)). \\
\begin{description}
    \item[Stable]: Does the motion remain bounded?
    \item[Unstable]: Does the motion grow unbounded?
    \item[Asymptotically stable]:Does the motion approach an equilibrium?
\end{description}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Consider the linear differential equation with constant real coefficients
\[\frac{d^n x}{dt^n} + b_1 \frac{d^{n-1}x}{dt^{n-1}} + \cdots + b_{n-1} \frac{dx}{dt} + b_n x = a_1 \frac{d^{n-1}u}{dt^{n-1}} + \cdots + a_{n-1} \frac{du}{dt} + a_n u\]
Equilibrium solution: \(x(t) \equiv 0\). \\
Taking the Laplace transform of the differential equation gives:
\[X(s) = \frac{a_1 s^{n-1} + \cdots + a_n}{s^n + b_1 s^{n-1} + \cdots + b_n} U(s)\]
Characteristic equation:
\[s^n + b_1 s^{n-1} + \cdots + b_n = 0\]
Let $p_i,i=1,2,\cdots,n$ be the solutions of the characteristic equation.
\begin{columns}
\column{0.3\textwidth}
\textcolor{blue}{Instability}:
\[\mathrm{Re}(p_i) > 0,\exists i\]
\vfill
\column{0.3\textwidth}
\textcolor{blue}{Asymptotic stability}:
\[\mathrm{Re}(p_i) < 0,\forall i\]
\vfill
\column{0.4\textwidth}
\textcolor{blue}{Stability}: \\
If none of the poles have positive real part, but there are distinct poles on the imaginary axis, then the system is stable
\end{columns}
\vfill
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin stabilization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Spin stabilization is one of the oldest forms of passive stabilization.
\begin{itemize}
    \item If the objective of a mission is to point a single instrument onboard the spacecraft in an inertially fixed direction, then we can spin the spacecraft about that axis.
\end{itemize}
\begin{center}\includegraphics{fig_10_1.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.1:} Spinning spacecraft\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Spin stability of torque-free motion:
\begin{itemize}
    \item We shall now examine the stability of spin equilibrium conditions.
    \item Let us consider the principal axes torque free Euler equations
    \[\begin{aligned}
    I_x \dot{\omega}_x + (I_z - I_y) \omega_y \omega_z &= 0 \\
    I_y \dot{\omega}_y + (I_x - I_z) \omega_z \omega_x &= 0 \\
    I_z \dot{\omega}_z + (I_y - I_x) \omega_x \omega_y &= 0
    \end{aligned}\]
    where \( I_x, I_y, I_z \) are called the principal moments of inertia.
    \item Equilibrium condition for the z-axis spin:
    \[ \omega_x(t) = \omega_y(t) = 0, \quad \omega_z(t) = v(\text{constant}) \]
\end{itemize}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{itemize}
    \item It is stable if either
    \[ I_z > I_y \text{ and } I_z > I_x \]
    or
    \[ I_z < I_y \text{ and } I_z < I_x \]
\end{itemize}
\begin{block}{Spin stability}
The spacecraft must be spinning either about the major (maximum) of minor (minimum) axis or inertia.
\end{block}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{columns}
\column{0.45\textwidth}
The preceding result relies on the fact that the body is rigid, and the motion is torque free.
\column{0.45\textwidth}
In practice, spacecraft are not entirely rigid, and the motion is not torque free (disturbance).
\end{columns}
\begin{columns}
\column{0.45\textwidth}
\column{0.45\textwidth}
\[\Downarrow\]
\textcolor{blue}{Internal Energy Dissipation}
\end{columns}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{columns}
\column{0.6\textwidth}
\begin{block}{Explorer 1}
\begin{itemize}
    \item This was the first U.S. satellite.
    \item It was axisymmetric, and spin-stabilized about the minor axis of inertia.
    \item Explorer 1 had flexible antennas, causing energy dissipation.
    \item The satellite began tumbling hours after deployment.
    \item This leads to the discovery of the major rule.
\end{itemize}
\end{block}
\column{0.3\textwidth}
\begin{center}\includegraphics[scale=0.5]{fig_10_2.jpg}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.2:} Explorer 1\end{center}
\end{columns}
\begin{block}{Major axis rule}
Spins about the major axis are asymptotically stable. Spins about any other axis are unstable.
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dual-spin stabilization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Spin stabilization is not always practical.
\begin{itemize}
\item The major axis rule requires that the spacecraft be spinning about the major axis,
    which may not be consistent with the shape of the spacecraft.
\item It means that the entire spacecraft is spinning. This becomes problematic in instances
    when it is necessary to point instruments in inertially changing directions.
    \begin{itemize}
    \item[\mysquare] For example, in telecommunication missions when the antennas
        are required to point toward the Earth, which is not an inertially fixed direction, but rotates once per orbit.
    \end{itemize}
\end{itemize}
\begin{center}\includegraphics[scale=0.6]{fig_6_1.jpg}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.3:} Communications satellite\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
It would be useful to have the benefits of spin stabilization (the gyroscopic stability), without having to spin the entire spacecraft, or at least without having to spin about a major axis.
\begin{block}{Dual-spin stabilization}
To accomplish this, a spinning wheel is mounted in the spacecraft with its spin axis aligned with the desired axis.
\end{block}
\begin{center}\includegraphics{fig_10_4.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.4:} Dual-spin spacecraft\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
    Effect of Internal Energy Dissipation
\begin{itemize}
    \item It is shown that it is possible to stabilize the major, minor and intermediate axis spins for a dual-spin spacecraft.
    \item But some energy dissipation is advantageous since the energy decay means that
    \[
    \vec{\omega} \to
    \begin{bmatrix}
    0 \\ 0 \\ \omega_z
    \end{bmatrix}
    \]
    which means \textcolor{blue}{it is safer to use the major axis as the rotation axis.}
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gravity-gradient stabilization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
There are a number of satellite applications that require the pointing of an instrument toward the Earth.
\begin{center}\includegraphics{fig_10_5.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.5:} Orbiting reference frame definition\end{center}
Let us define \textcolor{blue}{an orbiting reference frame} $\mathcal{F}_o$
 with basic vectors \( \vec{x}_o, \, \vec{y}_o, \, \vec{z}_o \) as follows
\[\vec{x}_o = \vec{y}_o \times \vec{z}_o, \, \vec{y}_o = -\frac{\vec{r} \times \vec{v}}{|\vec{r} \times \vec{v}|}, \, \vec{z}_o = -\frac{\vec{r}}{|\vec{r}|}\]
where \( \vec{r} \) and \( \vec{v} \) are the spacecraft position and velocity relative to the center of the Earth respectively.
\end{frame}

\begin{frame}[t]{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\textcolor{blue}{
It should be clear that if the spacecraft body frame \( \mathcal{F}_b \) is aligned with the orbiting frame \( \mathcal{F}_o \), then the body z-axis points directly to nadir as desired for an Earth pointing satellite.}
\vfill
\begin{center}\includegraphics{fig_10_5.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.6:} Orbiting reference frame definition\end{center}
\vfill
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Let us consider the movement under the gravity-gradient torque
\[T_g = \frac{3\mu}{r^5} r_b^\times Ir_b\]
\vspace{-12pt}
\begin{block}{}
The spacecraft attitude motion relative to the orbiting frame $\mathcal{F}_o$ is stable
 for small attitude deviations from $\mathcal{F}_o$ if $\mathcal{F}_b$ is a principal axis frame,
 and the principal inertias satisfy
\vspace{-6pt}
\[I_y > I_x > I_z\]
\end{block}
\vspace{-6pt}
\begin{center}\includegraphics{fig_10_5.pdf}\end{center}
\vspace{-6pt}
\begin{center}\textcolor{blue}{Figure \arabic{section}.7:} Orbiting reference frame definition\end{center}
\end{frame}
